| import OAI.Analysis.IrrationalRotation.Model.AffiliatedTransport |
| import Mathlib.Analysis.VonNeumannAlgebra.Basic |
|
|
| namespace OAI |
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| noncomputable section |
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|
| namespace InvariantRotation.AffiliatedTransport |
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|
| variable {𝕜 E : Type*} [RCLike 𝕜] |
| variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] |
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|
| @[simp] |
| theorem mem_domainImage {H : E →ₗ.[𝕜] E} {p : Submodule 𝕜 E} {y : E} : |
| y ∈ domainImage H p ↔ ∃ x : H.domain, (x : E) ∈ p ∧ H x = y := by |
| simp [domainImage, Submodule.mem_map] |
|
|
| theorem domainImage_invariant |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) (A B : E →L[𝕜] E) |
| (hAp : ∀ x ∈ p, A x ∈ p) |
| (hAD : ∀ x ∈ H.domain, A x ∈ H.domain) |
| (hAB : ∀ x : H.domain, H ⟨A x, hAD x x.property⟩ = B (H x)) : |
| ∀ y ∈ domainImage H p, B y ∈ domainImage H p := by |
| intro y hy |
| obtain ⟨x, hxp, rfl⟩ := mem_domainImage.mp hy |
| exact mem_domainImage.mpr |
| ⟨⟨A x, hAD x x.property⟩, hAp x hxp, hAB x⟩ |
|
|
| theorem transported_invariant |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) (A B : E →L[𝕜] E) |
| (hAp : ∀ x ∈ p, A x ∈ p) |
| (hAD : ∀ x ∈ H.domain, A x ∈ H.domain) |
| (hAB : ∀ x : H.domain, H ⟨A x, hAD x x.property⟩ = B (H x)) : |
| ∀ y ∈ transported H p, B y ∈ transported H p := by |
| have hi : domainImage H p ∈ Module.End.invtSubmodule B.toLinearMap := |
| (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr |
| (domainImage_invariant H p A B hAp hAD hAB) |
| exact (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp |
| (Submodule.topologicalClosure_mem_invtSubmodule hi) |
|
|
| theorem transported_invariant_of_commutes |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) (W : E →L[𝕜] E) |
| (hWp : ∀ x ∈ p, W x ∈ p) |
| (hWD : ∀ x ∈ H.domain, W x ∈ H.domain) |
| (hWH : ∀ x : H.domain, H ⟨W x, hWD x x.property⟩ = W (H x)) : |
| ∀ y ∈ transported H p, W y ∈ transported H p := |
| transported_invariant H p W W hWp hWD hWH |
|
|
| theorem mem_transported_orthogonal |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) |
| (hH : H.IsFormalAdjoint H) (x : H.domain) (hx : H x ∈ pᗮ) : |
| (x : E) ∈ (transported H p)ᗮ := by |
| rw [transported, Submodule.orthogonal_closure, Submodule.mem_orthogonal] |
| intro y hy |
| obtain ⟨z, hzp, rfl⟩ := mem_domainImage.mp hy |
| rw [hH z x] |
| exact p.inner_right_of_mem_orthogonal hzp hx |
|
|
| theorem inverse_domainImage_le_orthogonal |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) |
| (hH : H.IsFormalAdjoint H) (hker : H.toFun.ker = ⊥) : |
| domainImage H.inverse pᗮ ≤ (transported H p)ᗮ := by |
| intro y hy |
| obtain ⟨z, hzp, rfl⟩ := mem_domainImage.mp hy |
| have hz : (z : E) ∈ LinearMap.range H.toFun := by |
| simpa only [LinearPMap.inverse_domain] using z.property |
| obtain ⟨x, hx⟩ := LinearMap.mem_range.mp hz |
| have hpartialker : H.ker = ⊥ := by simp [LinearPMap.ker, hker] |
| rw [LinearPMap.inverse_apply_eq hpartialker hx] |
| apply mem_transported_orthogonal H p hH x |
| change H.toFun x ∈ pᗮ |
| rw [hx] |
| exact hzp |
|
|
| theorem inverse_transported_le_orthogonal |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) |
| (hH : H.IsFormalAdjoint H) (hker : H.toFun.ker = ⊥) : |
| transported H.inverse pᗮ ≤ (transported H p)ᗮ := |
| Submodule.topologicalClosure_minimal _ |
| (inverse_domainImage_le_orthogonal H p hH hker) |
| (Submodule.isClosed_orthogonal _) |
|
|
| section Complete |
|
|
| variable [CompleteSpace E] |
|
|
| theorem inverse_transported_le_orthogonal_of_selfAdjoint |
| (H : E →ₗ.[𝕜] E) (p : Submodule 𝕜 E) |
| (hH : IsSelfAdjoint H) (hker : H.toFun.ker = ⊥) : |
| transported H.inverse pᗮ ≤ (transported H p)ᗮ := by |
| have hs := LinearPMap.adjoint_isFormalAdjoint hH.dense_domain |
| rw [LinearPMap.isSelfAdjoint_def.mp hH] at hs |
| exact inverse_transported_le_orthogonal H p hs hker |
|
|
| end Complete |
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|
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| variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] |
|
|
| theorem transported_projection_mem |
| (M : VonNeumannAlgebra F) (H : F →ₗ.[ℂ] F) (p : Submodule ℂ F) |
| [p.HasOrthogonalProjection] (hp : p.starProjection ∈ M) |
| (hD : ∀ W ∈ M.commutant, ∀ x ∈ H.domain, W x ∈ H.domain) |
| (hcomm : ∀ W, ∀ hW : W ∈ M.commutant, ∀ x : H.domain, |
| H ⟨W x, hD W hW x x.property⟩ = W (H x)) : |
| (transported H p).starProjection ∈ M := by |
| apply (VonNeumannAlgebra.IsStarProjection.mem_iff |
| isStarProjection_starProjection M).mpr |
| intro W hW |
| rw [Submodule.range_starProjection] |
| apply (Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mpr |
| have hinv := (VonNeumannAlgebra.IsStarProjection.mem_iff |
| isStarProjection_starProjection M).mp hp W hW |
| rw [Submodule.range_starProjection] at hinv |
| exact transported_invariant_of_commutes H p W |
| ((Module.End.mem_invtSubmodule_iff_forall_mem_of_mem _).mp hinv) |
| (hD W hW) (hcomm W hW) |
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|
|
| end InvariantRotation.AffiliatedTransport |
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|
| end |
|
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| end OAI |
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|
